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Many optimization problems can be cast into the maximum satisfiability(MAX-SAT) form, and many solvers have been developed for tackling suchproblems. To evaluate a MAX-SAT solver, it is convenient to generate hardMAX-SAT instances with known solutions. Here, we propose a method of generatingweighted MAX-2-SAT instances inspired by the frustrated-loop algorithm used bythe quantum annealing community. We extend the algorithm for instances ofgeneral bipartite couplings, with the associated optimization problem being theminimization of the restricted Boltzmann machine (RBM) energy over the nodalvalues, which is useful for effectively pre-training the RBM. The hardness ofthe generated instances can be tuned through a central parameter known as thefrustration index. Two versions of the algorithm are presented: the random- andstructured-loop algorithms. For the random-loop algorithm, we provide athorough theoretical and empirical analysis on its mathematical properties fromthe perspective of frustration, and observe empirically a double phasetransition behavior in the hardness scaling behavior driven by the frustrationindex. For the structured-loop algorithm, we show that it offers an improvementin hardness over the random-loop algorithm in the regime of high loop density,with the variation of hardness tunable through the concentration of frustratedweights.

Generating Weighted MAX-2-SAT Instances of Tunable Difficulty with Frustrated Loops